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228 IX ANALYTIC FUNCTIONS

Observe that this last result is not true for analytic functions of real
variables (1/(1 + x²) is a counterexample). Also, a continuous function
/(X y) of two real variables may be analytic in each of the variables without
being analytic in R²; an example is given by f(x, y) = xy²/(x² + y²) for

Remark. It follows from (9.9.4) that the set F, product of the circles
\x_k — a_k\ = r_k (1 < k ^p) is a set of uniqueness in A (when A is connected);
for the power series (c_v(z — a)^v) is entirely determined by the values of/
on F, hence if two analytic functions in A coincide in F, they coincide in P,
and the result follows from (9.4.2).

PROBLEMS

1. Let A be a relatively compact open connected set in C. Let 99 be a continuous mapping
of [a, b] x [0, 1] into A such that t -> <p(t, f) = y_?(/) is a circuit contained in A for
0<£<1, and t-+yo(t) = <p(t₉Q) is a circuit contained in A (which may contain
frontier points of A). Suppose in addition that for every e > 0, there exists S > 0 such
that the relation |A — /x| < 8 implies |yi(/) — y'n(t)\ ^ e for / e [a, b] — D, where D is a
denumerable subset.

Let now / be a continuous mapping of A into a complex Banach space E, such

that its restriction to A is analytic. Show that Cauchy's theorem f(z) dz = I f(z) dz

Jyo Jyi

still holds (use (8.7.8)).

2. Let A be an open subset of C, /a continuous mapping of A into a complex Banach
space E, such that/is analytic in A n D+ and A n D_, where D₊ (resp D_) is defined
by ./(z) > 0 (resp ./(z) < 0). Show that /is analytic in A. (Suppose the disk \z\ < r is
contained in A. Let y+ (resp. y_) be the circuit defined in [— 1, -f 1] by y+(t) = (2f + l)r
for -1 < t < 0, y+(0 = re^nit for 0 ^ /< 1 (resp. y_(/) = re^nil for -1 ^ t < 0,
y_(/) = (1 - 2t)r for 0 ^ / < 1.) Show that if \z\ < r and ./(z) > 0, then

2iriJy+ x — z⁹ 2m Jy- x—z

using Problem 1; hence if y is the circuit t-^re^nit in [—1, 4-1],

1 f f(x) dx

2m Jy x — z

Then use (9.9.2).)

3. Show that the conclusion of (9.9.4) still holds when/is merely assumed to be bounded
in each bounded polydisk contained in A, but not necessarily continuous. (Use Prob-
lem 6 of Section 8.9; actually, a deep theorem of Hartogs shows that even this weakened
assumption is not necessary; in other words, a function which is analytic separately
with respect to each of the;? complex variables z, is analytic in A.)



	228 IX ANALYTIC FUNCTIONS Observe that this last result is not true for analytic functions of real variables (1/(1 + x²) is a counterexample). Also, a continuous function /(X y) of two real variables may be analytic in each of the variables without being analytic in R²; an example is given by f(x, y) = xy²/(x² + y²) for

	Remark. It follows from (9.9.4) that the set F, product of the circles \x_k — a_k\ = r_k (1 < k ^p) is a set of uniqueness in A (when A is connected); for the power series (c_v(z — a)^v) is entirely determined by the values of/ on F, hence if two analytic functions in A coincide in F, they coincide in P, and the result follows from (9.4.2).

	PROBLEMS 1. Let A be a relatively compact open connected set in C. Let 99 be a continuous mapping of [a, b] x [0, 1] into A such that t -> <p(t, f) = y_?(/) is a circuit contained in A for 0<£<1, and t-+yo(t) = <p(t₉Q) is a circuit contained in A (which may contain frontier points of A). Suppose in addition that for every e > 0, there exists S > 0 such that the relation \|A — /x\| < 8 implies \|yi(/) — y'n(t)\ ^ e for / e [a, b] — D, where D is a denumerable subset. Let now / be a continuous mapping of A into a complex Banach space E, such that its restriction to A is analytic. Show that Cauchy's theorem f(z) dz = I f(z) dz Jyo Jyi still holds (use (8.7.8)). 2. Let A be an open subset of C, /a continuous mapping of A into a complex Banach space E, such that/is analytic in A n D+ and A n D_, where D₊ (resp D_) is defined by ./(z) > 0 (resp ./(z) < 0). Show that /is analytic in A. (Suppose the disk \z\ < r is contained in A. Let y+ (resp. y_) be the circuit defined in [— 1, -f 1] by y+(t) = (2f + l)r for -1 < t < 0, y+(0 = re^nit for 0 ^ /< 1 (resp. y_(/) = re^nil for -1 ^ t < 0, y_(/) = (1 - 2t)r for 0 ^ / < 1.) Show that if \z\ < r and ./(z) > 0, then

	2iriJy+ x — z⁹ 2m Jy- x—z using Problem 1; hence if y is the circuit t-^re^nit in [—1, 4-1], 1 f f(x) dx 2m Jy x — z Then use (9.9.2).) 3. Show that the conclusion of (9.9.4) still holds when/is merely assumed to be bounded in each bounded polydisk contained in A, but not necessarily continuous. (Use Prob- lem 6 of Section 8.9; actually, a deep theorem of Hartogs shows that even this weakened assumption is not necessary; in other words, a function which is analytic separately with respect to each of the;? complex variables z, is analytic in A.)